On asymptotic formulae for solutions to differential equations with summable coefficients (Q2711768)

The authors suggest a method for the construction of asymptotic solutions to the linear system \(\dot x=A(t)x\) by reducing it to the so-called \(L\)-diagonal form. It is assumed that on \([t_0,\infty)\) the eigenvalues of matrix \(A(t)\) are multiple, the corresponding elementary divisors are of constant multiplicity, and the matrix \(D(t):=J(t)+T^{-1}(t)T'(t)\) has simple eigenvalues. Here, \(J(t)\) is the Jordan normal form of \(A(t)\) and \(T(t)\) is such a nonsingular matrix that \(T^{-1}(t)A(t)(t)T(t)=J(t)\). Thus after the change of variables \(x=T(t)y\) one obtains a system of the form \(\dot y=D(t)y\). By introducing an artificial small parameter \(\varepsilon \) the authors apply a method of asymptotic diagonalization to the system NEWLINE\[NEWLINE\varepsilon \dot y=D(t)y.\tag{1}NEWLINE\]NEWLINE More precisely, the matrix \(U_m(t, \varepsilon)=\sum_{s=0}^{m}\varepsilon^sU_s(t)\) is constructed in such a way that the change of variables \(y=U_m(t,\varepsilon)z\) transforms (1) to the system NEWLINE\[NEWLINE\varepsilon \dot z=(\Lambda_m(t,\varepsilon)+\varepsilon ^{m+1}C_ m(t,\varepsilon))z\tag{2}NEWLINE\]NEWLINE with the diagonal matrix \(\Lambda_m(t,\varepsilon) \sum_{s=0}^{m}\varepsilon^s\Lambda_s(t)\). Under the assumptions that for \(\varepsilon =1\) the eigenvalues of this matrix have different real parts on \([ t_0,\infty)\) and \(\int_{t_0}^{\infty }\|C_m(t,1)\|dt<\infty \), the authors apply well known results from the theory of \(L\)-diagonal systems [see, e.g. \textit{R. Bellman}, Stability theory of differential equations. New York-London: McGraw-Hill Book Company (1953; Zbl 0053.24705)] to construct solutions to system (1) for \(\varepsilon =1\).NEWLINENEWLINEFor the entire collection see [Zbl 0937.00046].